Global Geometry of planar vector fields

平面矢量场的全局几何

• 批准号: RGPIN-2015-04558
• 负责人: Schlomiuk, Dana
• 金额: $ 0.8万
• 依托单位: Université de Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=664476
• 关键词: Global; Geometry; planar; vector; fields

This proposal is on planar polynomial vector fields.  There are several hard problems on these systems  still unsolved after more than a century: two problems posed by Poincaré and one posed by Hilbert, the second part of his 16th problem. The proposed research is expected to throw light on these open problems and will also have an impact on applications.  The following points indicate the main research lines of this project.  ***I) Global studies of families of polynomial vector fields.  The class of quadratic differential systems is 5-dimensional modulo the action of the group of affine transformations and time rescaling.  So far only sub-families of dimension at most three have been studied from a truly global viewpoint, among them the 3-dimensional family of Lotka-Volterra systems, very important for applications, studied by the applicant together with N. Vulpe. Some other important sub-families will be studied, among them the 4-dimensional family of quadratic vector fields with a first order weak focus, important  for Hilbert's 16th problem. Some cubic families will also be studied.***II) The applicant will continue her study with her collaborators on the classification of singularities of the whole quadratic class. Together with Artés, Llibre and Vulpe she already published 4 papers on this topic, another paper has been accepted for publication and another one was submitted in July 2014.  There remain 6 cases to be studied and form part of this research proposal: the case with four distinct real finite singularities, with 4 distinct complex singularities, with two real and two complex singularities, with 3  real singularities, one of them double, with two complex simple and one real double singularities and with a single real singularity of multiplicity four. Over 1000 global configurations of singularities for the quadratic class are expected. This rather huge classification theorem will have applications. Indeed, in the study of models in  applied mathematics involving quadratic systems, our algorithm could be used for effective computation of the global configurations of the singularities of the systems involved, information  available at the touch of a button.***III) Another line of research is on problems of integrability of polynomial vector fields in terms of invariant algebraic curves. Poincaré's problem on algebraic integrability and problems resulting from the work of Darboux on integrability in terms of algebraic invariant curves are open.  The Theorem of Darboux (or of Jouanolou) gives only sufficient conditions for Darboux (respectively algebraic) integrability and the number of curves involved is not optimal. The applicant wants to detect necessary conditions for algebraic or Darboux integrability, involving the degrees and multiplicities of the curves, conditions on the their relative position such as  intersection multiplicities of the curves and conditions on their singularities.   **

这个建议是关于平面多项式向量场的。 有几个困难的问题，这些系统仍然没有解决后，世纪：两个问题提出的庞加莱和一个提出的希尔伯特，第二部分，他的第16个问题。预计拟议的研究将揭示这些开放的问题，也将对应用产生影响。 以下几点表明了本项目的主要研究路线。 *I）多项式向量场族的全局研究。 一类二次微分系统是5维模仿射变换和时间重标度的作用组。 到目前为止，从真正的全局观点来看，仅研究了维数至多为3的子族，其中，对于应用非常重要的Lotka-Volterra系统的3维族由申请人与N.狐狸其他一些重要的子家庭将研究，其中4维家庭的二次向量场与一阶弱焦点，重要的希尔伯特的第16个问题。也将研究一些立方族。* II）申请人将继续与合作者一起研究整个二次类的奇点分类。与Artés，Llibre和Vulpe一起，她已经发表了4篇关于这个主题的论文，另一篇论文已被接受出版，另一篇论文于2014年7月提交。 还有6种情况需要研究，并构成本研究建议的一部分：具有4个不同的真实的有限奇点的情况，具有4个不同的复奇点的情况，具有两个真实的和两个复奇点的情况，具有3个真实的奇点的情况，其中一个是双奇点的情况，具有两个复单奇点和一个真实的双奇点的情况，以及具有一个多重数为4的真实的奇点的情况。超过1000个全球配置的奇点的二次类预计。这个相当庞大的分类定理将有应用。事实上，在研究涉及二次系统的应用数学模型时，我们的算法可以用于有效计算所涉及系统的奇点的全局配置，只需按一下按钮即可获得信息。III）另一条研究路线是关于多项式向量场在不变代数曲线方面的可积性问题。庞加莱问题的代数可积性和问题所产生的工作达布的可积性方面的代数不变曲线是开放的。 达布定理（或Jouanolou定理）只给出了达布（分别为代数）可积的充分条件，并且所涉及的曲线数不是最优的。申请人希望检测代数或达布可积性的必要条件，涉及曲线的次数和多重性，关于它们的相对位置的条件，例如曲线的相交多重性和关于它们的奇点的条件。 **